Keith Walker wrote:
>Moorad Alexanian writes:
>>I would not equate the notion of a change of c with time with that of a
>>change of the conversion factor between calories and joules.
>Quite right. Space-like and time-like dimensions in Minkovski space have
>different properties. We are not dealing with a standard Euclidean space
>here. This is indicated by the negative sign applied (solely) to the time
>dimension in the metric for space-time, and by the related fact that we
>speak of the arrow of time but not the arrow of space.
I disagree with Keith and Moorad, and agree with George here. It is true
that Lorentz signature effects give the time-like direction in spacetime a
different geometric character than the space-like directions. But this is
no reason to consider the time-like direction as having a different
engineering dimension than the space-like directions, and to measure them in
different units. Just which direction in spacetime constitutes the
time-like direction depends on the particular Lorentz frame used.
Different frames have their respective time direction pointing in somewhat
different directions. But different frames still refer to the *same*
underlying Minkowski spacetime manifold with only different coordinate
labels on the spacetime points (events). Since the time-direction in one
frame is a mixture of time and space directions in another frame it is
cumbersome to consider the time dimension as having a different
(engineering) dimension than the spatial ones -- even though it contributes
to the metric with an opposite sign.
Also it is possible to make an analytic continuation of the time direction
by the Wick rotation t --> i*t in the complex plane which then gives
spacetime a purely Euclidean character (but with complex dimensions). If a
quantity is represented by a complex number we usually do not represent its
imaginary part with a different engineering dimension than its real part.
For instance, consider the concept of an electrical impedence. Both the
real and imaginary parts have the dimension of J/(s*A^2) = 'ohm' even
though the imaginary reactance part behaves differently mathematically,
i.e. algebraically, than the real resistance part in calculations.
Also, because time has an arrow whereas space does not is not a valid
reason to give time-like spacetime intervals a different engineering
dimension than space-like spacetime intervals. The fact that time has an
arrow does not come from relativity. It comes from thermodynamics (and
also from the breakdown of CP invariance in certain elementary particle
interactions). These arrows have nothing to do with the local geometric
structure of spacetime. In addition, it is conceivable and some
observations suggest that it may be possible that space *does* have an arrow
as well. (See Nodland & Ralston, Phys. Rev. Lett., Apr. 21, 1997, "Is the
Universe Birefringent?".) The existence or absence of an arrow for
some directions of spacetime is a issue separate from whether or not
the value of the speed limit of causation c = 299792458 m/s acts solely in
the equations of physics as a unit conversion factor. It converts
spacetime intervals denominated in 'temporal' units to spacetime intervals
denominated in 'spatial' units, and is analogous to the unit conversion
factor in thermodynamics of 4.184 J/cal which converts between
energy changes denominated in 'heat' units and energy changes denominated
in 'work' units.
>Notwithstanding that I agree with Moorad about the inapplicability of
>George's argument at this point, I'm not convinced that there is any
>evidence that 'c' has fluctuated. So in terms of conclusions I'm with
>George and Glenn and co..
Here I agree with Keith, George, Glenn, & co.
P.S. I'd like to here Don Page's thoughts on this matter of whether or not
'c' is a conversion factor analogous to the relationship between joules and